Abstract. The complexity of Aq-eigenmaps, i.e. homogeneous degree q harmonic polynomial maps f: S '~ --+ S n, increases fast with the degree q and the source dimension m. Here we introduce a variety of methods of manufacturing new eigenmaps out of old ones. They include degree and source dimension raising operators. As a byproduct, we get estimates on the possible range dimensions of full eigenmaps and obtain a geometric insight of the harmonic product of A2-eigenmaps.
Mathematics Subject Classification (1991). 58E20.
1. Introduction and Preliminaries
It is well known that a spherical harmonic of order q, i.e. an eigenfunction of the Laplace-Beltrami operator A sm with eigenvalue S Aq = q(q + m - 1), is the restriction (to S m) of a homogeneous harmonic polynomial of degree q in m + 1 variables. A map f : S m --+ S v into the unit sphere of a Euclidean vector space V is said to be a Aq-e igenmap if all components of f are spherical harmonics of order q. A Aq-eigenmap is a harmonic map with constant energy density Aq/2 [1] and their 'classification' is a fundamental problem raised in [1]. Apart from the classical examples such as the Hopf map and the Veronese surfaces (or more generally, the standard minimal immersions), only a few explicit examples are known. The objective of this paper is to give various new constructions that give rise to a variety of new examples of eigenmaps between spheres. In Section 2 we define the degree raising operator that associates to a Aq-eigenmap f : S m --+ S n a Aq+l-eigenmap f+ : S m --+ S( m+0(~+t)-l . By iteration, this is then generalized to raising the degree by an arbitrary positive integer. In Section 3, we introduce the source dimension raising operator that associates to a A2-eigenmap f : S ra -+ S n
a A2-eigenmap f : S ~+1 --+ 5 '~+'~+2. We obtain, as a corollary, ten new range dimensions for full A2-eigenmaps for m >_ 5. The main result here (for the lowest range dimension) asserts rigidity of the Hopf map among A2-eigenmaps f : S m --+
58 HILLEL GAUCHMAN AND GABOR TOTH
S 2, m >_ 2. In Section 4 we define the harmonic product of two eigenmaps and show that this includes the degree raising operator discussed in Section 2. The main result of this section is a general existence theorem of the harmonic product for A2-eigenmaps. Finally, in Section 5, this result is used to give examples and nonexamples for the harmonic product.
2. Raising the Degree
Let H denote the harmonic projection operator [6]. H is the orthogonal projection from the vector space ~q of homogeneous polynomials in m + 1 variables of degree q onto the linear subspace of harmonic polynomials of the same degree (cf. Vilenkin [6]).
Let f: S m ---+ Sv be a Aq-eigenmap. We define
f+: Rm+ 1 _, R(m+l)(n+ 1)
as follows. The components of f+ are given in double indices i = 0 , . . . , m and j = 0 , . . . , n b y
( f+ )i=J c+H(x i f f )
where
/2q + ra - 1 = i
Our first result (proved in [5]) asserts that f+ is a Aq+l-eigenmap. Since the developments of the rest of this section depend on this claim, for completeness, we give here the details.
LEMMA 1. f+ maps the unit sphere to the unit sphere so that the restriction f+: S m ~ S (m+l)(n+l)- I is a Aq+l-eigenmap.
Proof. By the harmonic projection formula [6] (or elementary computation), and harmonicity of the components i f , we have
p2 Off H ( x i f J ) = x i f J 2 q + m - 1 0 x i '
where p2 m 2 f j = ~i=oXi. Homogeneity of has two consequences. First, we have
o f f i=o Xi Oxi = qfj"
Second, since f maps the unit sphere to the unit sphere, we have ~ = o ( f J ) 2 -- p2q as polynomials. Applying the Laplacian A m Rm+l m 2 2 = = ~i=o 0 /Ox i to both sides and restricting to S m, we obtain
~_, 2.., \ -~x i ] = q ( 2 q + m - 1 ) . i=0 j=O
HARMONIC POLYNOMIAL MAPS 59
Thus
H ( x i f J ) 2 = x i : j i=0 j=O i=0 j=O
1 Of j ~ 2 2q + r n - 1 0 x i )
= 1 2q + m - 1 Oxl
j=O i=0
+ (2q + 1)2 i=0 j=0
2q q = 1 2 q + r a _ l + 2 q + m _ ~
q + m - 1
2 q + m - 1
and the lemma follows. A map between spheres is said to be f u l l if the image is not contained in any
proper great sphere. Restricting the range to the least great sphere it is contained in, a nonfull map can always be made full. Notice that f+ is not full even if f is. In what follows we will denote a map and its full restriction by the same letter. Two maps f l : S m ---+ Sv1 and f2: S m ---+ Sv2 are said to be equivalent if there exists an isometry A: V1 --+ V2 such that f2 = A o fl. After having f+ made full it is clear that fl and f2 equivalent implies that f+ and f+ are also equivalent. The converse, though much deeper, is also true and is proved in [5], namely, that the + operation is injective on the set of equivalence classes of ,~q-eigenmaps.
We now generalize this to raising the degree by an arbitrary positive integer p by iteration. In what follows, we use standard multiindex notation, namely a multiindex oe = ( ao , . . . , am) always has nonnegative integer components,
= m . ce! m = 1-Ii=o Let E PP, I a I = p, bethe [c~ I Ei=0c~,, = 1-I/=0 ai and x ~' m xi~,. e,~ homogeneous polynomial of degree p given by
e'~(x) = ~ x~
and define the scalar product in 7 )p such that {e~}l~l=p is an orthonormal basis. Given a Aq-eigenmap f: S m ~ S n, we define
f+,p: R m + 1 __.+ p P ® R n+l
by
(f+'P)~=Cp,qH(e a . f j ) , [ a l = p , j = O , . . . , n,
60 HILLEL GAUCHMAN AND GABOR TOTH
where
/ (2q+ m - 1)(2q+ m + 1) . . . (2q + m + 2 p - 3)
~" '~: V ~q¥ ~-1--~ ~ ~)_:7(q-~-~;--~-) THEOREM 1. f +'P maps the unit sphere to the unit sphere so that it restricts to a Ap+q-eigenmap f+'P: S m ~ Sr, p®R,+~.
Proof. We use induction with respect to p to show that
n
2 ~_~ ~ H(e,~fj)2 = p2(p+q) (1) ep,q
I~l=p j=O
Forp = 1, this is the statement of the lemma above noting that Cl,q : Ct" Assum- ing that (1) is true for all Aq-eigenmaps, we compute
n n
E E/~(~°s~) ~ ~ E E × = Cp÷l , q Cp÷ 1 ,q
Io4=p+, j=o M=p+I j=0
> ( p + l ) ! H(x~O...x~mfj) 2 ao{. . , a~{
m rb
: c~,,,q E E z~(X~o...x~mS~) 2 io,...,ir,=O j=O
m n m
= CpT , , q ' ' i=O j=O il,...,ip=O
771 n
2 Y~ E E H(xiH( e'~fj))2" : Cp÷l, q i=0 j=0 I~l=p
By the induction hypothesis, c,,qH(eC~f j) are components of a Ap+q-eigenmap. Hence, by Lemma 1,
Combining this with the result above and noting that Cp+l,q = Cp,qC++q the theorem follows.
3. Raising the Source Dimension
To manufacture new Aq-eigenmaps out of old ones, the degree raising operation discussed in the previous section allows us to restrict ourselves to q = 2. As, for
HARMONIC POLYNOMIAL MAPS 61
m = 3, a complete classification of ~2-eigenmaps is known (cf. [3]), we now describe an operation on )~2-eigenmaps that raises the source dimension. In fact, given a )~2-eigenmap f : 5"m __+ Sn, we define
f : R m+2 ___+ Rn+m+3
by
f ( x ) = 1+ r e ( m + 2 ) f (x ) , m+l m + 2 '
X O X m + I ~ • . . ~ ~- - - X m X m + I m
x = ( x 0 , . . ,
PROPOSITION 1. Given a A2-eigenmap f: S m --+ S n, the induced map f maps the unit sphere to the unit sphere so that it restricts to a A2-eigenmap f : 5'm+1 ~ S~+m+2. Moreover, f full implies that f is full.
Proof. Simple computation.
The maximum range dimension for a full A2-eigenmap f : S TM --+ 5'~ is
m ( m + 3) 1 2
that is the multiplicity of the eigenvalue )~2 minus one. For m = 3, the possible range dimensions of full ,~2-eigenmaps f : 5'3 ~ 5',~ are n = 2, 4, 5, 6, 7, 8. Combining these with Proposition 1, we obtain the following:
C O R O L L A R Y 1. For m >_ 3, there exist full )~2-eigenmaps
f : s m --+ 5 " ( m ( r a + 3 ) / 2 ) - r
where
r = 1 , 2 , 3 , 4 , 5 , 7 .
For example, it fol lows that, for m = 4, full ),2-eigenmaps f : S 4 --+ S '~ exist for n = 7, 9, 10, 11, 12, 13. Moreover, n = 4 can be added to this list since the gradient o f an isoparametric function gives a full )~2-eigenmap. Note also that, by a result o f R. Wood (el. [7]), there is no full polynomial map f : 5 '4 --+ S 3 so that n = 3 does not arise as a range dimension.
Remark. Precomposing these with the Hopf map h: S 7 ~ 5 '4, we obtain )~4- eigenmaps f : S 7 ~ 5'~, where n = 4, 7, 9, 10, 11, 12, 13.
Let F : R m × R TM ~ R n be an orthogonal multiplication, i.e. F is bilinear and
62 HILLEL GAUCHMAN AND GABOR TOTH
[ F ( z , y) [ = I z I I Y l, z, y E R m. The Hopf-Whitehead construction associates to F the )~2-eigenmap
f F : S 2ra- 1 ----r ~n
defined by
f F ( z , y) = (I x 12 -- l Y 12 , 2F(x , y)).
Clearly, fF is full iff F is onto. Note that, leaving harmonicity, a general result of R. Wood in [7] asserts that any quadratic polynomial map f between spheres is homotopic to an f r associated to an orthogonal multiplication F: R z × R TM --+ R n. Returning to the harmonic case (l = m) above, for m = 2, the possible range dimensions are n -- 2 and 4 corresponding to the complex multiplication and the real tensor product (cf. [4] for further details). By [2], for m = 3, the possible range dimensions are n - 4, 7, 8 and 9 (the first corresponding to quaternionic multiplication). Combining these range dimensions with the ones obtained from Corollary 1 (for m = 5), it follows that there exist full )~2-eigenmaps f: S 5 S n for n = 4, 7, 8, 9, 13, 15, 16, 17, 18, 19. For example, in contrast to the nonexistence of full )~2-eigenmaps f : S 4 ---+ S 3, the lowest range dimension here gives a full )~2-eigenmap f : S 5 ~ S 4. In fact, restricting the quaternionic multi- plication to a pair of three-dimensional subspaces gives a full orthogonal multipli- cation F: R 3 × R 3 --+ R 4 and the Hopf-Whitehead construction provides a full A2-eigenmap fF: S 5 --+ S 4. Raising the source dimension as above, we obtain the following:
COROLLARY 2. For m >_ 5, there exist fuU )~2-eigenmaps
f: S TM -.+ s(m(m+3)/z)-r,
where
r = 1, 2, 3, 4, 5, 7, 11, 12, 13, 16.
Finally, note that there is no orthogonal multiplication F: R 3 × R 3 --+ R 3 so that the Hopf-Whitehead construction does not give any ,~z-eigenmaps f : S 5 ~ S 3.
As for the minimum range dimension, we have the following rigidity result:
THEOREM 2. Let f: S m ~ S z be a fuU )~2-eigenmap. Then m = 3 and, up to isometries on the source and the range, f is the Hopf map.
Remarks. 1. Hence the only unsettled range dimensions of a full )~z-eigenmap f : S 4 ---+ S n are n = 5, 6 and 8.
2. For any degree q, we can define
N(q) -- min{n I there exists afull )~q-eigenmap
f : S TM --+ S ~ for some ra > 2}
HARMONIC POLYNOMIAL MAPS 63
Clearly, N(q) <_ 2q, for any q (because of the Veronese map f : S 2 ---+ S 2q) but, for q even, N(q) <_ q as composition of the Hopf map and the Veronese map shows.
PROOF OF THEOREM 2. Let f: S m ~ S n be a Az-eigenmap. Using coordinates, we write
m
f ( x ) = E aix2 + E i=0 O<i<j<m
a i j x i x j ,
whereai , aij E R n+l, i = 0 , . . . , m a n d 0 _< i < j _< m.To simplify the notation, we set aij = aft, so that aij is defined for all distinct indices 0 _< i, j _< m. Harmonicity is equivalent to
m
E ai=O. i=0
(2)
Writing out the condition that f maps the unit sphere into the unit sphere, we obtain the following:
l ai I = 1, (3)
(ai, aij) = O, i, j distinct, (4)
[ aij ] 2 -t- 2(ai, aj) = 2, i, j distinct, (5)
(ai, ajk) + (aij, aik) = O, i, j, k distinct, (6)
(aij , akl) + (aik, aft) + (air, ajk} = O, i, j, k, I distinct. (7)
We say that a system of vectors {ai, aij} C R n+l is feasible if it satisfies (2)-(7). A feasible system is generic if ai ~ q-aj for all i, j distinct. We first show that by precomposing f with a suitable isometry on the source, the associated feasible system of vectors can be mad e generic (provided that m _> 2).
Let 0 <_ r < s < rn and consider the rotation in the xTx,-plane by angle 4. Denoting the new coordinates by the superscript 4, we obtain
x~ = x~ cos 4 + xs sin4,
x ¢ = -x~ sin 4 + xs cos 4,
and the rest of the coordinates are unchanged. For the new system of vectors {a/~ } we have
ar ~ = ar COS2 4 -~- as s in2 ¢ - a~, sin 4 cos 4,
a~ = a~ sin e 4 + as cos 2 4 -~- ars sin 4 cos 4,
64 HILLEL GAUCHMAN AND GABOR TOTH
with the rest of the vectors unchanged. Clearly, ar + as = ar + as. Since the ai's are unit vectors, we have the following three cases:
Case 1: ar and as are linearly independent. By (3)-(5), at , as and ars span a three-dimensional subspace in which we have rotation around the axis a~ + as by angle 2¢.
Case 2: a~ q- as. Everything stays fixed. Case 3: ar = - a s . By (5), [ars [ = 2 and so ar and ars/2 is an orthonormal
basis in the plane they span. The opposite pair of vectors ar c and a~ is obtained from a~ and as by rotation with angle 2¢ in this plane.
We now prove the claim about making a system of feasible vectors generic. Given a feasible system of vectors, assume that ai = aj, for some i ~ j . By (2), there exists ak that is different from these. We can now rotate ai and ak corresponding to Case 1 or Case 3 to get three vectors such that each two are linearly independent. In this way we can decrease the number of identical vectors without increasing the number of opposite pairs. Since the number of vectors is finite we arrive at a feasible system of vectors in which the vectors ai are all distinct. Assume now that ai = - a j . Since m _> 2 there exists ak that is linearly independent from these. We now rotate ai and ak as in Case 1 to get all three linearly independent without creating new identical pairs. After finitely many steps, we arrive at a generic system.
We now assume that n = 2 and denote the vector cross product in R 3 by x. By (3)-(5), we have
aij = eij l + (al, aj) ai X aj, i • j , (8)
where eij = + 1 is antisymmetric in i j . We now claim that any three distinct vectors ai, aj and ak span R 3. This is clear, since otherwise aij, ajk and aki were parallel, contradicting (5) and (6).
We claim that, for any distinct indices i, j and k, we have
(ai, aj) + (aj, ak) + (ak, ai) = --1. (9)
Letting
#ij = (ai, aj) ,
we substitute (8) into (6) and obtain
2 X ak) + (a. ejk 1 + #j-------7
aj
2 (ai x aj, ai x ak) = O. + eijeik
~/(1 + Ply)(1 + #ik)
HARMONIC POLYNOMIAL MAPS 65
On the other hand, we have
(ai X aj , ai X ak) = (aj , ak) - (ai, ak)(ai , aj)
= #jk -- #i j#ik
so that
i 1 + #jk (a~, aj x ak) = *qcjkCk~v~ (1 + Vq)(1 + ~k) ( , jk - ~ , ~ k ) , (10)
where we used antisymmetry of the E's. The left-hand side is the (signed) volume of the parallelepiped spanned by ai, aj and ak. We now take a cyclic permutation of i, j and k and note that Eijedkeki does not change. We obtain
(1 + #jk)(#jk - V;j~k~) = (1 + #kd(Vk; - ~;j~jk),
or equivalently
( l + #ij + # j k )# j k = (1 + #ij + #ki)#ki . (11)
Taking a cyclic permutation of i, j and k and subtracting it from (11), we arrive at
(1 + #~j + #jk + #kd(mj - #k~) = 0.
Thus either (9) holds or #ij = #ik. Taking a cyclic permutation again and noting that (9) remains invariant, it remains to study the case when
(ai, a j ) = (aj , ak) = (al~, ai). (12)
To finish the proof of the claim, we now show that this implies (ai, a j) = - 1 / 3 . For this, we compute the volume of the parallelepiped spanned by ai, aj and ak. We obtain that the volume is (1 - #)2(1 + 2#), where # is the common value of (12). Substituting this into (10), we arrive at 1 + 3# = 0 and (9) follows.
Finally, we add a new vector at to ai, aj and ak. (Note that at exists since m _> 3.) By the above, (9) is valid with i, j or k replaced by I. Using these four equations, we obtain that
(a i + aj + ak + al, ai + aj) = 0 .
Similarly
(ai + aj + ak + al, ak + at) = O.
Adding these, we obtain
ai + aj + ak + al = O.
Thus, at is uniquely determined by ai, aj and ak and using that the system is genetic we get m = 4. By [3], any )~2-eigenmap f : S 3 ~ S 2 is, up to isometries
66 HILLEL GAUCHMAN AND GABOR TOTH
on the source and the range, the Hopf map and the proof is complete.
Remark. Given an orthogonal multiplication F: R m x R m ~ R n, setting x = ( x l , . . . , Xm) and y = (Xm+l , . . . , X2m), we can write
rn 2m
F(x, y)= E E aijxixj. i=O j=mT1
The vectors a i j C R n satisfy the relations
[ aij [ = 2,
(aij, aik) = 0, i, j , kdistinct,
and (7), ,#here we assume that aij : aji and that aij is zero if 1 _< i, j _< ra or m + 1 _< i, j _< 2ra. Hence fF corresponds to the feasible system of vectors {ai, aij} C R n+l, where al = . . . = am = - a m + l = " ' - = -a2m being orthogonal to all aij's.
4. The Harmonic Product
We first reformulate Theorem 1 in a more convenient setting. Let
f)~v: R m + l "-+ 73v
be the harmonic polynomial map given by
A.(x) : cp,0 H(e )(x)e [~[=p
where the constant is given before Theorem 1. The proof of Theorem 1 shows that f~v maps the unit sphere into the unit sphere so that it restricts to a Ap-eigenmap f;~v: Sm --+ Spy.
Let 7-/p denote the space of spherical harmonics of order p on Sm. We endow 7~ p with the normalized scalar product
n(A.) + 1 /8 'v (g' g ' ) - vol(S ) m g 9 s
where
n(Ap) + 1 = dim 7-/p
and vsm is the volume form on S '~ with total volume vol(Sm). Given an orthonor-
mal basis {f~p} C 7-/p, we define the standard minimal immersion [4]
fay: S m __+ Sn(;~p)
HARMONIC POLYNOMIAL MAPS 67
by
n(~p) A , ( x ) = 5 . ( x ) S p , x c s m
j=O
THEOREM 3. f;~,, when made full, is equivalent to the standard minimal immersion
f~,. Proof. This is a simple consequence of the fact that an equivariant eigenmap
is standard. Equivariance of f;~v can be seen easily since the harmonic projection operator commutes with the actions of SO(m + 1) on 7 )p and on 7-/p given by precomposition with the inverse. This latter is because H is built from the powers of the Laplace-Beltrami operator.
We now introduce a law of composition for eigenmaps. Let f : S m ~ Sv be a Ap-eigenmap and g: S ra ~ S w a Aq-eigenmap. Their tensor product f ® g: S ~ x S ~ ~ Sv®w is a Ap+q-eigenmap. (For the general properties of the ten- sor product of eigenmaps, cf. [4].) Restricting to the diagonal S ~ C S m x S ~ is, in general, not an eigenmap and so, instead, we think of f @ g as a har- monic polynomial map of R m+l x R m+l to V @ W and restrict to the dia- g o n a l R m+l C R ra+l × R m+l to get a polynomial map ( f@g) [Rm+l : R m+l V ® W. Finally we take the harmonic part of each component with respect to an(y) orthonormal basis to get a harmonic polynomial map I I ( ( f ® g) [R-~+I): Rm+l --+ V @ W. If it maps the unit sphere into the unit sphere (up to a constant multiple) then its restriction is called the harmonic product of f and g denoted by f 0 9 : S m ~ Sv®w. We now show that Theorem 1 can be reinterpreted in terms of the harmonic product as follows:
THEOREM 4. For any Aq-eigenmap f: S ra ~ S n the harmonic product f~v <) f exists.
For the proof, we need the following:
LEMMA 2. For g E 7-{ p+I and9 ~ E 7{ p, we have
Og g '} = #p(g, H(xig'))p+l,
where
2 p + m - 1 #p = ( p + 1) p ~ - ~ - 1
Proof. Simple computation (cf. also [5]).
PROOF OF THEOREM 4. Choosing an orthonormal basis in "~{P for the compo- nents of the standard minimal immersion f~v: S m --+ S ~(;~p) and using Lemma 2, we compute
68 HILLEL GAUCHMAN AND GABOR TOTH
j=O /=0 E ~ g ~ OXil .OXip Xil'''Xip fl j=O l=O ih...,ip=O " "
x H ( f~ . f ' )
X
_ 1 p ! - p ~ ~ ~ ~_~ ~ H(x~ ' f ' ) ×
j=o l=o I~l=v
( ) x H ~,E~-'--~,, ,m f~pfl v , , o 0 • • • v t o m
#o , . . tzp - l p!
~(.~p) ,~ p!
EEE × j=o l=0 I~l=p
X H(xC'fZ)H((fJ~p, H(xC~))fJ~pf t)
2 '~ P! =
t=0 I~l=v
Cp, 0 t=0 I•l=p
The proof of Theorem 1, however, shows that this is equal to
a Cp,O p2(p+q) C2 q
which completes the proof.
The harmonic product is clearly commutative and associative, i.e. with obvious notations, we have
f O(gOh) = ( f O g ) O h ,
provided that f 0g , g (> h and either side exist. In particular, by Theorem 4,
( f O g ) + = y+ Og = f Og + ,
where the equalities mean equivalence. Therefore to study the existence of the harmonic product we may assume that the factors are ),2-eigenmaps.
The rest of this section is devoted to the proof of a useful criterion for the existence of the harmonic product of ~2-eigenmaps. Let f : S "~ --+ S u and g: S m
HARMONIC POLYNOMIAL MAPS 69
j u l v (g)l=o' In 5 'v be •2-eigenmaps with components ( f ) j=0 and what follows, the indices i, k, r, s will mn on 0 , . . . , m while j (resp. l) will take values on 0 , . . . , u (resp. 0 , . . . , v). We now set
u 'o
fik = ~_, fJ 02fj j=o Oxi Oxk and gik = Z gl 02g l l=o Oxi OXk"
(Note that the partial derivatives are actually constants.)
THEOREM 5. Given )~2-eigenmaps f: S m ~ S ~ and g: S m -+ S ~, the harmonic product f 0 g exists iff
m fikgik = C" p4, C = constant. (13)
i,k=O
Proof. The harmonic projection of a degree 4 polynomial ¢ C p4 is given by
p2 p4
H ( ¢ ) = ¢ 2 ( m + 5 ) A ¢ + s ( m + 3 ) ( m + 5 ) A2¢
(cf. Vilenkin [6]). In particular, we have
, °2 ~ Of j Og l H(fJg l) f i g I +
2 ( m + 5 ) z~. Oxi Oxi
p4 ~-~ 02fJ 02gl +
2(m + 3)(m + 5) ~ Oxi Oxk Oxi Oxk"
Up to a constant multiple, this is the jl-component of the harmonic product. We claim that the norm square of the corresponding vector is
j,l +5 + c(f' g) p8 + 2 ( m + 3 ) ( m + 5 ) 2 x
X ( 4 ( m T 4 ) ~ f i k g i k - - p 2 A ( ~ i , k (14)
where the constant c(f, g) is given by
c(f~ g) 4(m + 3)2(m + 5) 2 j,l
To show this, we first decompose
02fJ 02g !
H(fJgl) 2 = A1 + A2 + A3 + B12 + B13 + B23, j,l
(~5)
70 H I L l , E L G A U C H M A N A N D G A B O R T O T H
where
A1 =
j,l
A2 (m + 5) 2 ~ O~c i Oz i '
_
A3 - 4( ra+3)2(m+ 5) 2 j~t
2P 2 fjgl Off Og t B12 = m + 5 ~ ~z/ Oxi' i,j,l
2 02fJ 02g l
O i-Sxk J
B13 p4 02fj 02gl
(m + 3)(m + 5) ~ fJgtoxi Ox-~k Oxi Oxk' i,j,k,l
B23 = p6 02gl
( m + 3 ) ( m + 5 ) ~ Off Og t 02ff i,j,t,r,s Oxi Ox~ Ox~. Oxs OxT Ox,"
We now simplify each term in a tedious but straightforward manner. It is clear that A1 = p8 since f and g are both )~2-eigenmaps between spheres. As for A2, we compute
02p 4 Op 4
We now make use of the fact that 02p 4
-- 4~ikp 2 q- 8XiX k Ox~ Oxk
gik) •
along with (harmonicity and) homogeneity of the components of f and g:
02 fJ 02g l Ei,k xiXk Oxi Oxk -- 2fj and Ei,k xiXk Oxi Oxk - 2gl
to arrive at _ _ _ p4
A2 4p8 -1- E fikgik. m + 5 ( m + 5 ) 2 i,k
(16)
HARMONIC POLYNOMIAL MAPS 71
We now notice that A3 is a constant multiple of p8; in fact
A3 = c(f , g)p8
where c(f , g) is given in (14). Turning to the mixed terms, we compute
BI2 - m + 5 G G ]
= - 2 ( m + 5 )
8p 8
m + 5 "
By the very definition of fik and gik, we have
/94
B13 = (rn + 3)(m + 5) ~ fikgik. i,k
Finally, we have
(17)
(18)
(19)
(m + 1 2c ) pS. H(fjgl)2---- \ m + 5 + ( m + 3 ) (m+ 5) 2 + c ( f , g) (21)
j,l
Conversely, assume that the harmonic product exists, i.e. Pq,ltt(fJgt) 2 is a constant multiple of pS. By (14), this means that # = Ei,kfikgik satisfies the equation
4(m + 4)# - p2 A # = ctp 4, c t = constant. (22)
Taking A of both sides and using homogeneity of A#, we obtain that
A# = c" p2~
so that
and this can be seen by working out the right-hand side (and using harmonicity of fJ and gl once again). Putting (16)-(20) together, (14) follows.
We now turn to the proof of the theorem. First, assuming (13), we have
B23 = 2(m + 3)(m + 5) 2 A fikgik (20)
72 HILLEL GAUCHMAN AND GABOR TOTH
where
c" = A2# + 4(m + 3)d m + 3
Finally, substituting this back to (22), we get
c t + c" p4 # - 4(m + 4)
and the proof is complete.
The normalizing constant for the harmonic product can be determined explicitly in terms of c in (13) by the following:
COROLLARY 3. Assume that (13) holds. Then
j,t + 5 + 2(m + 3)(m + 5) pS. (23)
Proof. Taking A 2 of both sides of (13), we obtain
Substituting this into (21) we arrive at (23). We now specialize to the case when f = g that is we turn to the existence of
the harmonic square f (> f. Let f: S m ~ S ~ be a full A2-eigenmap and assume that the harmonic square f ~ f exists. By Theorem 5, we then have Ei,kf2k = cp 2 so that (by restriction)
]: S ~ __+ S~(~+2)
HARMONIC POLYNOMIAL MAPS 73
defined by
1 f = - ~ (fik)i,~=O
is clearly a A2-eigenmap.
THEOREM 6. ], when rnade full, is equivalent to f .
Before the proof we need the following:
LEMMA 3. span{fik}~,k=O = span{fJ}~= 0.
Proof. Since the fik are linear combinations of i f , we have span{fik} C span{if}. Assuming that the inclusion is proper, let a: span{if} --* R be a nonzero linear functional that vanishes on span{f/k}. Setting c d = a(fJ), we have
Ozi Oxk = ~" cj Oxi Oxk" J
Hence
cj 02 f j ~,j,~ Ox~ Oz-------~k z~xk = 2 ~j cjfJ = o
that is a contradiction since f is full and not all the cj are zero.
PROOF OF THEOREM 5. Let
A: R n+l --+ R (re+l)2
be the linear map defined by
1 Ouf j 3 ( y ) = ~ 0x~ 0xk yj , y = (yJ)~=0 e rt n+l
i,k=O
Then, we have ] = A o f . By Lemma 3, the image V of A is of dimension n + 1. Hence A is a linear isomorphism between R n+l and V and, by assumption on the existence of the harmonic square, it maps the unit sphere into the unit sphere. Thus, A is an isometry and the proof is complete.
5. Existence and Nonexistence of the Harmonic Product
5.1. THE GENERALIZED HOPF MAP
Using complex coordinates zo , . . . , Zm E C m+l, the generalized Hopf map
h: S 2m+l -+ S (3ra(ra-1-1)/2)-I
74 HILLEL GAUCHMAN AND GABOR TOTH
is defined by
h(zo,..., zm)=
1 2 ( ~ 1)~(z,~k)) o_<,<k_<,,,
h is clearly a &2-eigenmap. For future reference, we note that, setting zi = xi + VcZ-f xT, i = 0 , . . . , ra and~ = r a + 1 , . . . , 2 r a + 1, we have
h. = h~ - 2(m + 1) ( d + 4 ) 2 m m + 3 p2, (24)
hik = h;~ - 2(m + 1) (xixk + x?xk), i • k, (25) m
hi ~ = _hlk _ 2(m + 1) ( x i x i - xkx;). (26) m
Let f : S 2m+1 --+ S'* be a ~2-eigenmap. We now recall that a real spherical harmonic ¢ of order 2 on S 2m+1 (such as fJ), written as a harmonic polynomial in the complex variables zi and ~i decomposes as
,;b - ¢11 -I- ~ x ,
where the 'pure part' ¢11 is the sum of those monomials in ¢ that are of degree 2 in zi or degree 2 in 2i and the 'mixed part' 4)× contains those monomials in ¢ that are of degree 1 in zi and degree 1 in zi. (Note that ¢11 (resp. Cx) are often refered as the (2, 0) + (0, 2)-part (resp. the (1, 1)-part) of ¢.) Clearly, ¢11 and C× are both harmonic.
We now view f as a vector-valued function whose components are spherical harmonics of order 2 and decompose accordingly:
f : y, + f× into pure and mixed parts, where fll, fx : S2m+] ~ t tn+l are vector-valued functions. It is natural to ask when will these map into the unit sphere, i.e. when will the pure and mixed parts of a )~2-eigenmap be again ~2-eigenmaps.
THEOREM 7. The harmonic product h 0 f o f the generalized Hopf map h: S 2 m + l --'-> S (3m(m+l)/2)-I and a ~2-eigenmap f: S 2m+1 ~ S ~ exists i f fboth the pure and mixed parts o f f are ~2-eigenmaps.
Proof. We use criterion (13) in Theorem 5 for the existence of h 0 f . By (24)-- (26), we have
HARMONIC POLYNOMIAL MAPS 75
2m+l ( 2 ( m ~ 1) 2 ) ho~so~ = ~ - (x~ + x~) m +---5 / ( s . + & ) +
a,b=O i
+ E (2( _m~ 1)(xixk + xTxT:)(fik + f;~)+ i¢k
+ 2 ( ~ + 1) ( .~ .~ _ .kx~)k~ + m
+ 2(ram Or 1) ( x k x ; - zix~)f;k)
2(m Or 1) 2m+l 2(m Or 1) 2m+l - m ~ Sobxox~ + ~ So,~yoyb, m a,b=O a,b=O
where we set Yi = -x: , Y7 = xi (and the p2 term cancels because of harmonicity). For the two terms on the right-hand side, with obvious notations, we have
E fabXaXb = E f j 02fj a,b a,b,j OXa OXb XaXb = 2 Ej (fj)2 = 2p2,
E fabyayb E fJ 02fj = - - - 2 ~ f J ( x ) f J ( y ) . ,,b a,b,j Oy,~ Oyb j
Putting these together, we obtain
2m+lE habfab- 4(m Or 1) p4 Or 4(m Or 1) ~ f j ( x ) f J (y ) - - m m a,b=O j
_ 4(m Or 1) p4 Or 4(m Or 1) ~ fj(z)fj(~--£-- ~ z), m m m j
where we used that yi Or v/:- f YT, = - x ; Or x/Z--1 xi = ~ zi and i f ( z ) means fJ written in the z-variables, etc.
We now apply Theorem 5 to conclude that h 0 f exists iff
fJ (z ) f f (x /Z- l z) = 0a. p4, w = constant. (27) J
Given a spherical harmonic ¢ of order 2 on S 2re+l, we have ¢× (v/-L-1 z) = ¢ × ( z ) and ¢11(~/7T z) = - e l l ( Z ) so that we have ¢(z)¢(v/-Z-1 - z) = ( ¢ × ( z ) Or ¢ll(Z))(¢x (z) - ¢11 (z)). Applying this to the left-hand side of (27), we obtain
It is clear that v is a ),2-eigenmap. The following result is the exact analogue of Theorem 7 and hence the proof is omitted.
THEOREM 8. The harmonic product v 0 f of the complex Veronese map v : S 2m+1 ---+
S(m+l)(m+2) -1 and a A2-eigenmap f: S 2m+l ~ S n exists iffboth the pure and mixed parts o f f are Az-eigenmaps.
In particular, since h is mixed and v is pure, we obtain that h 0 h, h <) v and v 0 v exist. Note that, for ra = 1 the )~2-eigenmaps of S 3 have been classified (cf. [3]) and from that further examples can be easily obtained.
5.3. THE o'-EXTENSION OF A ,~2-EIGENMAP
To obtain examples for nonexistence of the harmonic product we first generalize the method of raising the source dimension of Section 3.
Let f : S m --+ S n be a A2-eigenmap and define
f : R m+2 __+ RN+I
by
H A R M O N I C POLYNOMIAL MAPS 77
( ( f ( z ) A f ( x ) , B 2 Cxixm+l , = Xm+l ra + 2 '
D x~ m + 2 , Dx~xs , O<_i,r,s<_m;r#s
2 and A, B, C, D are constants with A # 0. Simple where p2 = x02 + . . . + Xm+l
computation shows that f maps the unit sphere into the unit sphere iff there exists an angle cr satisfying
m + l cos 2 a > - - (29)
m + 2
such that
A - (m + 2)v m + 1
m + 1 V ~°s 2 cr 2' m +
m + 2 B - - - c o s (7,
m + l
~/2(ra__+ 2) C = V r a + 1 ,
r a + 2 D = - - sin or.
+ 1
Given cr satisfying (29), the restriction f ° = f: S m+l --+ S u defined above is called the cr-extension o f f . (Note that N = n + 2m + 3 + ra(ra + 1)/2.) Clearly, cr = 0 reduces to the case treated in Section 3.
THEOREM 9. Let f : S m --+ S u and g: S m --+ S v be Az-eigenmaps and cr and such that
COS 20"~ COS 2 0 > - - r a + l
r a + 2 '
Assume that f ~ g exists. Then the harmonic product fa 0 9 o of the extensions does not exist.
and similarly for g. In what follows we use (13) of Theorem 5 to conclude the nonexistence of f f 0 g °. By assumption, we certainly have (13). By elementary (but long) calculation, we obtain
m+l
E a,b=0
a O 4 2 2 f~kgik = KP 4 + LXm+l + MXm+lP ,
where
L 2 2 ( 4 ( m + 1)~ 4( ra+ 1) 2 2 2 2
= A fAg \c + ] + (AfDa + AaDI) + m m
2 2 4B}B 2 8 ( m + 2 ) 2 + 4DIDa + (m + 1) 2 '
( 8 ( m + 1) ) 2 2 8 ( m + 1) 2 2 2 2 2 M = - 2C+m2(m+2) AIAa m ( m + 2 ) ( A I D g + A a D ] ) -
8 2 2 8(m -~ 1) 2 2 8(m -1- 2) 2 ra+2 BfBg r a + 2 DfDg+ ( r e + l ) 2 '
where the constants corresponding to f and g are indicated by subscripts. (The corresponding expression for K is irrelevant.) Assume now that f¢ 0 gO exists. Then, by Theorem 5, L = M = 0. In particular, we have
0 = 2 L + M = ( m D } + ( r a + 1)AEf)(mD 2 +(ra+ 1)32 ) +
2 2 m2(m + 2)3 (30) + m2( + 1)BsB. (m + 1)2
We now use the actual values of A, B, C, D to obtain mD 2 + (m + 1)A 2 = m(m + 2)/(m + 1). Substituting this into (30), we get B}B 2 = (m + 2)2/(m + 1) 2 or, equivalently, cos 2 cr cos 2 0 = (m + 1)2/(m + 2) 2 which clearly contradicts the assumptions.
HARMONIC POLYNOMIAL MAPS 79
References
1. Eells, J. and Lemaire, L.: Selected topics in harmonic maps, Reg. Conf. Ser. in Math., No. 50, AMS, 1982.
2. Parker, M.: Orthogonal multiplications in small dimensions, Bull. London Math. Soc. 15 (1983), 368-372.
3. Toth, G.: Classification of quadratic harmonic maps of S 3 into spheres, Indiana Univ. Math. J. 36(2) (1987), 231-239.
4. Toth, G.: Harmonic Maps and Minimal Immersions through Representation Theory, Academic Press, Boston, 1990.
5. Toth, G.: Mappings of moduli spaces for harmonic eigenmaps and minimal immersions between spheres, J. Math. Soc. Japan44(2) (1992), 179-198.
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